Some Einstein nilmanifolds with skew torsion arising in CR geometry

Abstract

We describe some new examples of nilmanifolds admitting an Einstein with skew torsion invariant Riemannian metric. These are affine CR quadrics, whose CR structure is preserved by the characteristic connection

Moduli spaces of type $$\mathcal {B}$$ B surfaces with torsion

“…There are, however, natural situations in which manifolds with torsion enter. We refer, for example, to work on torsion-gravity [4,11,20,21,22,36,37,45], on hyper-Kähler with torsion supersymmetric sigma models [23,24,25,43], on string theory [29,32], on almost hypercomplex geometries [38], on spin geometries [33], on B-metrics [39,44], on contact geometries [1], on almost product manifolds [41], on non-integrable geometries [2,8], on the non-commutative residue for manifolds with boundary [46], on Hermitian and anti-Hermitian geometry [40], on CR geometry [17], on Einstein-Weyl gravity at the linearized level [16], on Yang-Mills flow with torsion [30], on ESK theories [14], on double field theory [31], on BRST theory [26], and on the symplectic and elliptic geometries of gravity [12].…”

Moduli spaces of oriented Type A manifolds of dimension at least 3

“…Although much of Riemannian geometry involves the study of the Levi-Civita connection, which is without torsion, in recent years connections which have torsion have played an important role in many developments. We refer, for example, to work on B-metrics [28,38,47,53], on almost hypercomplex geometries [46], on string theory [1,27,33,37], on spin geometries [39], on torsion-gravity [5,12,19,20,21,22,44,45,54], on contact geometries [2,28], on almost product manifolds [49], non-integrable geometries [3,9], on the non-commutative residue for manifolds with boundary [55], on Hermitian and anti-Hermitian geometry [48], CR geometry [17], hyper-Kähler with torsion supersymmetric sigma models [23,25,24,52], Einstein-Weyl gravity at the linearized level [16], Yang-Mills flow with torsion [34], ESK theories [14], double field theory [36], BRST theory [26], and the symplectic and elliptic geometries of gravity [13]. Perhaps surprisingly, even the 2-dimensional case is of interest; connections on surfaces have been used to construct new examples of pseudo-Riemannian metrics without a corresponding Riemannian counterpart [10,11,15,…”